\chapter{Some Basic Properties of Bessel Functions}\label{chap:Besselfunctions}
Below, we collect some important properties of Bessel functions that have been heavily used in solving the radiation problem with a cylindrical boundary condition. 
Detailed properties of Bessel functions may be found in Ref.~\cite{Watson1995}.

Differential equations define Bessel functions are in the following general form,
\begin{align}
x^2\sdd{R(x)}{x}+x\dd{R(x)}{x}+(x^2-m^2)R(x)=0,
\end{align}
where $R(x)$ is a Bessel function with index $m$. 

Recurrence relations for the first kind of Bessel functions:
\begin{align}
J_m(x)=\frac{m+1}{x}J_{m+1}(x)+\dd{J_{m+1}(x)}{x}=\frac{m-1}{x}J_{m-1}-\dd{J_{m-1}(x)}{x}.
\end{align}

Derivatives of Bessel functions:
\begin{align}
J_m^\prime(x)&=\frac{1}{2}(J_{m-1}(x)-J_{m+1}(x))\\
{H^{(1)}_m}^\prime (x) &=\frac{1}{2}({H^{(1)}_{m-1}}^\prime (x)-{H^{(1)}_{m+1}}^\prime(x)).
\end{align}